Question: Antonio's toy boat is bobbing in the water next to a dock. Antonio starts his stopwatch, and measures the vertical distance from the dock to the height of the boat's mast, which varies in a periodic way that can be modeled approximately by a trigonometric function. The vertical distance from the dock to the boat's mast reaches its highest value of $-27 \text{ cm}$ every $3$ seconds. The first time it reaches its highest point is after $1.3$ seconds. Its lowest value is $-44\text{ cm}$. Find the formula of the trigonometric function that models the vertical height $H$ between the dock and the boat's mast $t$ seconds after Antonio starts his stopwatch. Define the function using radians. $ H(t) = $ What is the vertical distance $2.5$ seconds after Antonio starts his stopwatch? Round your answer, if necessary, to two decimal places. $ $
Answer: Let's start by writing a formula for the height of the boat's mast $u$ seconds after its peak. Both sine and cosine can be used to model periodic contexts. We can decide which is better fitting by considering the $y$ -intercept. The sine function intercepts the $y$ -axis at its midline, and the cosine function intercepts the $y$ -axis at its peak. The boat reaches its highest point at $u = 0$, so let's use a cosine function. The amplitude of boat's height function is $\dfrac{-27 - (-44)}{2} = 8.5\text{ cm}$. The period is $3$ seconds, since a cosine function reaches its peak once in every period. The midline is the average of the highest and lowest values, or $\dfrac{(-27) + (-44)}{2} = -35.5$ Since the ordinary cosine function $f(u) = \cos u$ has period $2\pi$, midline $y = 0$, and amplitude $1$, we need to stretch it horizontally by a factor of ${\dfrac{3}{2\pi}}$, stretch it vertically by a factor of ${8.5}$, and move it down ${35.5}$ units: $ H(u) = {8.5}\cos\left({\dfrac{2\pi}{3}}u\right) - {35.5}$ Since the boat's first peak is after $1.3$ seconds, a time that's $t$ seconds after Antonio starts his stopwatch is $t - 1.3$ seconds after the peak. So $u = t - 1.3$ : $ H(t) = {8.5}\cos\left({\dfrac{2\pi}{3}}(t - 1.3)\right) - {35.5}$ After $2.5$ seconds, the height of the boat is $\begin{aligned} H(2.5) &= 8.5\cos\left(\dfrac{2\pi}{3}(2.5-1.3)\right) - 35.5\\ &\approx -42.38\end{aligned}$ A correct formula for $H(t)$ is: $ H(t) = 8.5\cos\left(\dfrac{2\pi}{3}(t - 1.3)\right) - 35.5$ The vertical distance after $2.5$ seconds is: $ -42.38\text{ cm}$